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Transportation Problem

A PresentationByProf. Manidatta Ray

Commodities are transported while people travel

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What is Transportation Problem ?

The Transportation Problem (TP)is a special

type of linear programming problem where theobjective is to minimize transportation cost i.e. , the

cost of distributing a product from a number ofsources or origins to a number of destinations .

In other words , TP is to transport variousamount of a single homogenous commodity , thatare initially stored at various origins , to differentdestinations in such a way that the total

transportation cost is a minimum .

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Example

Lets consider a manufacturer who operates 3factories and dispatches his products to 5 differentretail outlets .

The table-1 indicates the capacities of the 3factories , the quantity of products required atvarious retail outlets and the cost of shipping oneunit of the product from each of 3 factories to each

of the 5 retail shops :

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Transportation Table :

FactoriesRetail Shops

Capacity

1 2 3 4 5

1 1 9 13 36 51 50

2 24 12 16 20 1 100

3 14 33 1 23 26 150

Requirements 100 70 50 40 40 300

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Terminologies :

The Origin of a transportation problem is the locationfrom which shipments are dispatched . The Destination of a transportation problem is the

location to which shipments are transported.

The table which provides the basic data regarding thetransportation problem is known as Transportationtable.

The quantities inside the colored rectangle are known asUnit Transportation Cost (UTC). It is the cost of

transporting one unit of the consignment from an origin toa destination .

A transportation problem is said to be unbalancedTP ifthe total availability at the origin is different from the total

requirement at the destination .

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Then the Transportation problem can beformulated as :

Minimize Z = C11X11+C12X12++C35X35 .Subject to Constraints:X11+X12+X13+X14+X15=a1.

X21+X22+X23+X24+X25=a2.

X31+X32+X33+X34+X35=a3.x11+x21+x31=b1.x12+x22+x32=b2.x13+x23+x33=b3.x14+x24+x34=b4.

x15+x25+x35=b5.andx11>=0, x12>=0,,x35>=0 .

This problem has 8 constraints and 15 variables .

The VariableXij indicatesthe amount to be transported

From the i th origin to j th destination

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Techniques for Solving TP

A no. of techniques are available for computingan initial basic feasible solution of atransportation problem :

North West Corner Rule .

Matrix Minimum Method/Least Cost Method .

VogelsApproximation Method .

But the optimum solution of a TP can be

obtained from : Modified Distribution Method (MODI Method) .

Stepping Stone Method .

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Thank You